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Properties

Label 806.2.bv.a
Level $806$
Weight $2$
Character orbit 806.bv
Analytic conductor $6.436$
Analytic rank $0$
Dimension $288$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [806,2,Mod(95,806)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(806, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([5, 24]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("806.95");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 806 = 2 \cdot 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 806.bv (of order \(30\), degree \(8\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.43594240292\)
Analytic rank: \(0\)
Dimension: \(288\)
Relative dimension: \(36\) over \(\Q(\zeta_{30})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 288 q - 36 q^{4} + 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 288 q - 36 q^{4} + 32 q^{9} + 18 q^{11} + 2 q^{13} + 12 q^{14} + 36 q^{16} + 10 q^{17} + 12 q^{19} - 18 q^{23} - 208 q^{25} - 24 q^{26} + 36 q^{27} - 32 q^{30} - 78 q^{33} + 32 q^{35} + 128 q^{36} - 12 q^{37} - 16 q^{38} - 48 q^{39} + 12 q^{41} + 12 q^{42} + 22 q^{43} + 60 q^{45} + 60 q^{46} - 22 q^{49} - 44 q^{51} + 10 q^{52} + 72 q^{53} - 36 q^{54} - 24 q^{55} - 4 q^{56} + 12 q^{58} + 42 q^{59} + 52 q^{61} - 38 q^{62} + 240 q^{63} + 72 q^{64} - 88 q^{65} - 96 q^{66} - 24 q^{67} + 40 q^{68} - 30 q^{69} - 12 q^{71} + 24 q^{72} - 4 q^{74} + 56 q^{75} - 18 q^{76} - 80 q^{77} + 132 q^{78} - 44 q^{79} + 32 q^{81} - 24 q^{84} - 72 q^{85} - 12 q^{87} - 20 q^{88} - 54 q^{89} + 36 q^{90} + 120 q^{91} - 16 q^{92} - 150 q^{93} + 12 q^{94} + 28 q^{95} - 60 q^{97} - 24 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
95.1 −0.406737 − 0.913545i −0.330997 − 3.14922i −0.669131 + 0.743145i 1.18603i −2.74233 + 1.58329i −2.94361 − 2.65044i 0.951057 + 0.309017i −6.87362 + 1.46103i 1.08349 − 0.482402i
95.2 −0.406737 − 0.913545i −0.288600 − 2.74584i −0.669131 + 0.743145i 1.00565i −2.39107 + 1.38048i 3.35011 + 3.01645i 0.951057 + 0.309017i −4.52191 + 0.961162i 0.918709 − 0.409036i
95.3 −0.406737 − 0.913545i −0.284200 − 2.70398i −0.669131 + 0.743145i − 2.34878i −2.35462 + 1.35944i 1.27596 + 1.14888i 0.951057 + 0.309017i −4.29630 + 0.913208i −2.14572 + 0.955336i
95.4 −0.406737 − 0.913545i −0.235845 − 2.24392i −0.669131 + 0.743145i 1.37728i −1.95400 + 1.12814i 1.08671 + 0.978475i 0.951057 + 0.309017i −2.04511 + 0.434701i 1.25821 − 0.560190i
95.5 −0.406737 − 0.913545i −0.162384 − 1.54498i −0.669131 + 0.743145i 2.67937i −1.34536 + 0.776744i −2.12233 − 1.91096i 0.951057 + 0.309017i 0.573853 − 0.121976i 2.44773 − 1.08980i
95.6 −0.406737 − 0.913545i −0.117564 − 1.11855i −0.669131 + 0.743145i − 0.854531i −0.974024 + 0.562353i −0.853015 − 0.768058i 0.951057 + 0.309017i 1.69712 − 0.360734i −0.780653 + 0.347569i
95.7 −0.406737 − 0.913545i −0.0721798 − 0.686745i −0.669131 + 0.743145i − 4.26135i −0.598015 + 0.345264i −2.52400 − 2.27262i 0.951057 + 0.309017i 2.46803 − 0.524597i −3.89293 + 1.73325i
95.8 −0.406737 − 0.913545i −0.0681106 − 0.648029i −0.669131 + 0.743145i 2.36130i −0.564301 + 0.325799i −0.337182 − 0.303600i 0.951057 + 0.309017i 2.51914 − 0.535460i 2.15716 − 0.960429i
95.9 −0.406737 − 0.913545i −0.0252910 − 0.240628i −0.669131 + 0.743145i − 3.19662i −0.209538 + 0.120977i 2.55107 + 2.29699i 0.951057 + 0.309017i 2.87718 − 0.611564i −2.92026 + 1.30018i
95.10 −0.406737 − 0.913545i −0.0133717 − 0.127223i −0.669131 + 0.743145i − 0.856178i −0.110785 + 0.0639618i 0.0499682 + 0.0449916i 0.951057 + 0.309017i 2.91844 − 0.620333i −0.782158 + 0.348239i
95.11 −0.406737 − 0.913545i 0.0692490 + 0.658861i −0.669131 + 0.743145i 3.80325i 0.573733 − 0.331245i 1.55296 + 1.39829i 0.951057 + 0.309017i 2.50514 − 0.532484i 3.47444 − 1.54692i
95.12 −0.406737 − 0.913545i 0.153009 + 1.45578i −0.669131 + 0.743145i − 0.0453058i 1.26769 − 0.731900i 2.27126 + 2.04505i 0.951057 + 0.309017i 0.838554 − 0.178240i −0.0413889 + 0.0184275i
95.13 −0.406737 − 0.913545i 0.176130 + 1.67576i −0.669131 + 0.743145i 3.04213i 1.45925 − 0.842498i 2.83612 + 2.55365i 0.951057 + 0.309017i 0.157278 − 0.0334304i 2.77912 − 1.23734i
95.14 −0.406737 − 0.913545i 0.182500 + 1.73637i −0.669131 + 0.743145i − 1.80835i 1.51202 − 0.872966i 0.592504 + 0.533493i 0.951057 + 0.309017i −0.0472232 + 0.0100376i −1.65201 + 0.735524i
95.15 −0.406737 − 0.913545i 0.201338 + 1.91560i −0.669131 + 0.743145i 0.383613i 1.66810 − 0.963078i −2.84267 − 2.55955i 0.951057 + 0.309017i −0.694559 + 0.147633i 0.350448 − 0.156030i
95.16 −0.406737 − 0.913545i 0.213100 + 2.02751i −0.669131 + 0.743145i − 0.841829i 1.76555 − 1.01934i −2.85220 − 2.56813i 0.951057 + 0.309017i −1.13095 + 0.240392i −0.769049 + 0.342403i
95.17 −0.406737 − 0.913545i 0.280111 + 2.66508i −0.669131 + 0.743145i − 4.25087i 2.32074 − 1.33988i −0.207581 − 0.186907i 0.951057 + 0.309017i −4.08974 + 0.869302i −3.88337 + 1.72899i
95.18 −0.406737 − 0.913545i 0.323106 + 3.07414i −0.669131 + 0.743145i 2.62520i 2.67695 − 1.54554i −0.424781 − 0.382475i 0.951057 + 0.309017i −6.41153 + 1.36281i 2.39824 − 1.06777i
95.19 0.406737 + 0.913545i −0.334280 − 3.18046i −0.669131 + 0.743145i 2.32155i 2.76953 − 1.59899i 1.37501 + 1.23807i −0.951057 − 0.309017i −7.06914 + 1.50259i −2.12084 + 0.944260i
95.20 0.406737 + 0.913545i −0.283805 − 2.70023i −0.669131 + 0.743145i − 3.18106i 2.35135 − 1.35755i −2.85981 − 2.57498i −0.951057 − 0.309017i −4.27624 + 0.908942i 2.90604 − 1.29385i
See next 80 embeddings (of 288 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 95.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner
31.d even 5 1 inner
403.bs even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 806.2.bv.a ✓ 288
13.e even 6 1 inner 806.2.bv.a ✓ 288
31.d even 5 1 inner 806.2.bv.a ✓ 288
403.bs even 30 1 inner 806.2.bv.a ✓ 288
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
806.2.bv.a ✓ 288 1.a even 1 1 trivial
806.2.bv.a ✓ 288 13.e even 6 1 inner
806.2.bv.a ✓ 288 31.d even 5 1 inner
806.2.bv.a ✓ 288 403.bs even 30 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(806, [\chi])\).